This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lmodvsubadd

Description: Relationship between vector subtraction and addition. ( hvsubadd analog.) (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmod4.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lmod4.p ⊢ + = ( +g ‘ 𝑊 )
lmodvaddsub4.m ⊢ − = ( -g ‘ 𝑊 )
Assertion lmodvsubadd ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )

Proof

Step Hyp Ref Expression
1 lmod4.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lmod4.p ⊢ + = ( +g ‘ 𝑊 )
3 lmodvaddsub4.m ⊢ − = ( -g ‘ 𝑊 )
4 lmodabl ⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Abel )
5 1 2 3 ablsubadd ⊢ ( ( 𝑊 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )
6 4 5 sylan ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )